A300824 -id:A300824 - cp's OEIS frontend

A300825 Filter sequence combining A300823(n) and A300824(n).

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 9, 2, 11, 2, 12, 13, 14, 2, 15, 16, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 44, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 64, 67, 68, 69

Offset: 1

Antti Karttunen, Mar 14 2018

nonn

Restricted growth sequence transform of ordered pair [A300823(n), A300824(n)].

For all i, j: a(i) = a(j) => A051953(i) = A051953(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A051953, A244042, A300222, A300823, A300824.

Cf. also A293226, A300833.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
A300823(n) = -sumdiv(n,d,(dA244042(d));
A300222(n) = (n - A244042(n));
A300824(n) = -sumdiv(n,d,(dA300222(d));
Aux300825(n) = [A300823(n), A300824(n)];
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300825(n))),"b300825.txt");

A300222 In ternary (base-3) representation of n, replace 1's with 0's.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2, 0, 0, 2, 6, 6, 8, 18, 18, 20, 18, 18, 20, 24, 24, 26, 0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2, 0, 0, 2, 6, 6, 8, 18, 18, 20, 18, 18, 20, 24, 24, 26, 54, 54, 56, 54, 54, 56, 60, 60, 62, 54, 54, 56, 54, 54, 56, 60, 60, 62, 72, 72, 74, 72, 72, 74, 78, 78, 80, 0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2

Offset: 0

Antti Karttunen, Mar 14 2018

			For n=46, which in base-3 (A007089) is 1201, replacing 1's with 0's gives 200, and as that is base-3 representation of 18 (= 2*(3^2) + 0*(3^1) + 0*(3^0)), a(46) = 18.

Antti Karttunen, Table of n, a(n) for n = 0..59048

Cf. A004488, A005836, A007089, A244042, A289814.

Cf. A300822 (Moebius transform).

Array[FromDigits[IntegerDigits[#, 3] /. 1 -> 0, 3] &, 93, 0] (* Michael De Vlieger, Mar 17 2018 *)

A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
A300222(n) = (n - A244042(n));
\\ Or directly as:
A300222(n) = fromdigits(apply(x->(if (1==x, 0, x)), digits(n, 3)), 3);

a(n) = n - A244042(n) = 2*A244042(A004488(n)).
a(n) = 2*A005836(1+A289814(n)). [With the current starting offset 1 of A005836.]
a(n) = A300822(n) + A300824(n).

A300822 Möbius transform of A300222.

Original entry on oeis.org

0, 2, 0, -2, 2, 4, 6, 8, 0, -4, 2, -4, 0, -6, 4, -2, 8, 12, 18, 22, 12, 14, 20, 16, 22, 24, 0, 0, 2, -8, 0, -4, 4, -4, 0, -12, 0, -18, 0, -28, 2, -12, 6, -8, 12, -4, 20, -4, 12, -2, 16, 0, 26, 36, 50, 48, 36, 50, 56, 44, 60, 60, 36, 52, 54, 28, 54, 52, 40, 60, 62, 48, 72, 72, 44, 72, 66, 48, 78, 82, 0, -4, 2, 0, -10, -6, 4, -10, 8, -24, -6, -14, 0

Offset: 1

Antti Karttunen, Mar 14 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A300222, A300821, A300824.

Table[DivisorSum[n, MoebiusMu[n/#] FromDigits[IntegerDigits[#, 3] /. 1 -> 0, 3] &], {n, 93}] (* Michael De Vlieger, Mar 17 2018 *)

A300222(n) = fromdigits(apply(x->(if (1==x, 0, x)), digits(n, 3)), 3);
A300822(n) = sumdiv(n,d,moebius(n/d)*A300222(d));

a(n) = Sum_{d|n} moebius(n/d)*A300222(d).
a(n) = A000010(n) - A300821(n).
a(n) = A300222(n) - A300824(n).

A300823 Difference between A244042 and its Möbius transform.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 4, 3, 2, 1, 4, 1, 0, 5, 0, 1, 6, 1, 14, 3, 8, 1, 8, 3, 12, 9, 16, 1, 14, 1, 10, 11, 8, 3, 12, 1, 0, 15, -4, 1, 12, 1, 8, 15, 2, 1, 10, 1, 8, 11, 4, 1, 18, 11, 24, 3, 26, 1, 28, 1, 30, 9, 30, 15, 20, 1, 32, 5, 46, 1, 24, 1, 36, 7, 40, 9, 24, 1, 50, 27, 38, 1, 60, 11, 36, 29, 32, 1, 42, 13, 32, 33, 26, 3, 50, 1

Offset: 1

Antti Karttunen, Mar 14 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008683, A051953, A244042, A300821, A300824, A300825.

f[n_] := FromDigits[IntegerDigits[n, 3] /. 2 -> 0, 3]; Table[f@ n - DivisorSum[n, MoebiusMu[n/#] f@ # &], {n, 97}] (* Michael De Vlieger, Mar 17 2018 *)

A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
A300823(n) = -sumdiv(n,d,(dA244042(d));

a(n) = A244042(n) - A300821(n).
a(n) = -Sum_{d|n, dA008683(n/d)*A244042(d) = Sum_{d|n, dA300821(d).
a(n) = A051953(n) - A300824(n).

cp's OEIS Frontend

A300825 Filter sequence combining A300823(n) and A300824(n).

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A300222 In ternary (base-3) representation of n, replace 1's with 0's.

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A300822 Möbius transform of A300222.

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A300823 Difference between A244042 and its Möbius transform.

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